**Solve the following :**

Find the area of the region bounded by the parabola y^{2} = x and the line y = x in the first quadrant.

#### Solution

To obtain the points of intersection of the line and the parabola, we equate the values of x from both the equations.

∴ y^{2} = y

∴ y^{2} – y = 0

∴ y(y – 1) = 0

∴ y = 0 or y = 1

When y = 0, x = 0

When y = 1, x = 1

∴ the points of intersection are O(0, 0) and A(1, 1). Required area:area of the region OCABO

= area of the region OCADO – area of the region OBADO

Now, area of the region OCADO

= area under the parabola y^{2} = x i.e. y = `± sqrt(x)` (in the first quadrant) between x = 0 and x = 1

= `int_0^1 sqrt(x)*dx`

= `[(x^(3/2))/(3/2)]_0^1`

= `(2)/(3) xx (1 - 0)`

= `(2)/(3)`

Area of the region OBADO

= area under the line y = x between x 0 and x = 1

= `int_0^1x*dx`

=`[x^2/2]_0^1`

=`(1)/(2) - 0`

= `(2)/(3)`

∴ required area = `(2)/(3) - (1)/(2)`

= `(1)/(6)"sq unit"`.